Open SMS messages sent through the Chrome Extension do not get synced back to the phone's messaging app

I'll try to be clear about what I'm doing to produce this. I'm in a group chat with two other people. I will send a text message via the Join extension in Chrome, the message will send just fine, however, my phone will never display that I sent a message in the chat. Upon closing and reopening the Join extension window, the message I just sent is not listed there.